A summary on random variables, covering: Bernoulli random variable, binomial random variable, negative binomial random variable, geometric random variable, hypergeometric random variable, poisson random variable, uniform random variable, exponential distribution, gamma distribution, and normal dist...
A random variable is essentially a random number. Random variables are upper case letters
and they can take on various values depending on the random outcome of an experiment. If
the random variable is given by a capital letter, the corresponding small letter denotes the
various values that the random variable can take on. Because the random variable depend on
the random outcomes of the experiment given in , the random variable is a function of .
2.1 Discrete Random Variable
Discreet random variable: A random variable X is discreet if it can take on a finite (or infinite
and countable) number of values, usually integers.
Events are often expressed as random variables in statistic, and we have to calculate
probabilities for these events (probability of the random variable).
Probability mass function: Suppose a random variable X can take on the values
(these values are known as mass points). The probability mass function (or frequency
function) of X is defined as the function such that: or
.
Properties of p: i)
ii)
The probability mass function (frequency function) can be represented by a bar chart. The
height of each bar equals the probability of the corresponding value of the random variable.
Example 1
A coin is tossed 3 times.
Sample space:
i) Let the random variable denote the number of heads.
1
, The values (mass points) that the random variable can take on is .
The probability mass function is then:
The probability mass function (frequency function) is represented by a bar chart in
Rice, p.36.
ii) Let the random variable Y denote the number of tails.
The values (mass points) that the random variable can take on is .
The probability mass function is then:
iii) Let the random variable Z denote the number of heads minus the number of tails.
The values (mass points) that the random variable can take on is .
The probability mass function is then:
2
, Example 2
Two socks are selected at random from a drawer containing five brown socks and three
green socks. Let X be the random variable where X is the number of brown socks selected.
The values (mass points) that the random variable can take on is .
The probability mass function is then:
Example 3
Check whether the following function:
can serve as the frequency function of a discreet random variable.
and
Thus, the given function can serve as the frequency function of the random variable with
values {1,2,3,4,5}.
Cumulative distribution function (distribution function)
There are many problems in which it is of interest to know the probability that the value of a
random variable is less than or equal to some real numbers.
Suppose X is a discreet random variable. Define a and b such that , then
.
3
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